Ghana Input-Output Matrix Analysis

Which sectors pull the rest of Ghana's economy along? Using the 2015 Social Accounting Matrix (SAM) for Ghana, produced by the Ghana Statistical Service (GSS), ISSER and IFPRI, I compute standard inter-industry linkage indices for its 56 commodity sectors. Eleven sectors have both above-average backward and above-average forward linkages. Eight of them hold up in every specification I tried: electricity, petroleum, wholesale and retail trade, information and communication, real estate, business services, wood and paper, and sugar cane.

Agriculture splits in two. Crops such as maize, rice and cotton buy few inputs but sell much of their output to other industries, while the food-processing industries that buy from farms have some of the strongest backward linkages in the economy. The staple crops grown across northern Ghana, including cassava, groundnuts, roots and vegetables, are near the bottom on both measures. The last part of the post explains why that does not make them a low priority. The analysis is preliminary.

What linkages measure

Backward linkages capture a sector's pull on its suppliers. When the sector expands, how much additional output does it require from the rest of the economy, directly and through successive rounds of input purchases? Forward linkages capture a sector's push on its customers: how much of the economy's production relies on its output as an input.

How they are computed

A SAM records every transaction in the economy in a single year: between production activities, commodities, labour and capital, households, government, investment and the rest of the world. The 2015 Ghana SAM has 55 activities and 56 commodities. Each activity produces a single commodity, so I work with a 56-sector commodity-by-commodity table. "Other cereals" is entirely imported and has no domestic output. Three adjustments turn the SAM's transactions into an input-output table:

• Margins. Each commodity's trade and transport margin is removed from every purchase of it and re-booked as a purchase of trade (66.5%) and transport (33.5%) by the same buyer.
• Taxes. Export taxes are removed from exports. Import and sales taxes are removed from all uses in proportion.
• Imports. Imported inputs are netted out in proportion to each commodity's import share of total supply, so the coefficients measure purchases from domestic producers only.

Households, government, investment and exports are treated as final demand (an open model). With \(z_{ij}\) the domestic purchases of sector \(j\) from sector \(i\) and \(x_j\) the gross output of sector \(j\), the technical coefficients and the Leontief inverse are

\[ a_{ij} = \frac{z_{ij}}{x_j}, \qquad L = (I - A)^{-1} = [l_{ij}]. \]

Following Rasmussen (1956), the backward linkage index of sector \(j\) is its column sum of \(L\), normalised by the economy-wide average:

\[ BL_j = \frac{\tfrac{1}{n}\sum_i l_{ij}}{\tfrac{1}{n^2}\sum_i\sum_j l_{ij}}. \]

The forward linkage index uses the Ghosh (1958) inverse, built from output-allocation coefficients, which give the share of sector \(i\)'s output sold to sector \(j\):

\[ b_{ij} = \frac{z_{ij}}{x_i}, \qquad G = (I - B)^{-1} = [g_{ij}], \qquad FL_i = \frac{\tfrac{1}{n}\sum_j g_{ij}}{\tfrac{1}{n^2}\sum_i\sum_j g_{ij}}. \]

Both indices average exactly 1 across the 56 sectors, so a value above 1 means above the economy average, not "strong" in any absolute sense. Sectors above 1 on both are often called key sectors. The spectral radius of \(A\) is 0.38, and \(L\) and \(G\) have no negative entries.

Figure 1. Backward and forward linkage indices for 55 sectors with domestic output, Ghana 2015 (domestic coefficients). Dot area is proportional to the sector's share of gross output. The dashed lines mark the economy average (1.0). Hover over a dot for the sector name and values; the full list is in the table at the end of the post.

The 11 key sectors

Sector	Backward	Forward	Share of output
Wood and paper	1.43	1.24	1.4%
Electricity, gas and steam	1.30	1.26	3.4%
Petroleum	1.30	1.70	3.1%
Business services	1.29	1.07	4.3%
Clothing*	1.21	1.05	0.2%
Wholesale and retail trade	1.17	1.21	8.0%
Information and communication	1.17	1.13	3.3%
Sugar cane	1.06	1.53	<0.1%
Forestry*	1.02	1.39	1.5%
Real estate activities	1.02	1.38	2.4%
Finance and insurance*	1.02	1.82	3.2%

* Not key in every variant (total coefficients, or Leontief-inverse rows for the forward index). Together, the 11 sectors account for 31% of gross output.

Reading the results

The key sectors are mostly energy and business services. Petroleum (1.30 backward, 1.70 forward) and electricity (1.30, 1.26) are key in every specification. So are trade, communications, real estate and business services, which buy from many industries and sell to almost all of them. Finance has the highest forward index in the economy (1.82), but its backward index (1.02) is barely above average and it drops out when imports are not netted out. Transport sells widely (forward 1.34) but buys relatively little from domestic producers (backward 0.96), so it is not key. Construction, the largest sector at 11% of gross output, is below average on both, a reminder that size and linkage are different things. Two key sectors, sugar cane and clothing, are each under 0.5% of gross output.

Crops push forward; livestock pulls back. Six crops have above-average forward linkages but below-average backward ones: maize (forward 1.33), sorghum and millet (1.34), rice (1.20), other oilseeds (1.39), tobacco (1.37) and cotton (1.45). They buy few inputs but sell much of their output to other industries. Cattle (backward 1.20), poultry (1.21) and other livestock (1.18) are the reverse: they pull more on domestic suppliers per cedi of output than the average sector. Ten of the 22 agriculture, forestry and fishing sectors are below average on both, including cassava, groundnuts, other roots, vegetables and cocoa (0.73, 0.68). Cocoa's weak forward linkage reflects the fact that most of the crop is exported, and exports count as final demand. Agriculture's average backward index is 0.84, against 1.10 for the rest of the economy.

Food processing pulls hardest on its suppliers. Six of the eight food, beverage and tobacco processing sectors have above-average backward linkages, including sugar refining (1.39, second-highest in the economy), meat, fish and dairy (1.32), fats and oils (1.24) and grain milling (1.20). None is a key sector, because most processed food is sold to households rather than to other industries (forward indices 0.69–0.84). Expanding food processing therefore pulls on its suppliers, which in a food chain begin with farms.

Netting out imports matters. If imported inputs are counted as if they were domestic (total coefficients), chemicals, non-metal minerals, metals and machinery all become key sectors. Their apparent backward linkages run largely through imports. With domestic coefficients they fall just below the average.

Low linkages do not mean low priority

The five crops supported by the livelihood grants in the programme discussed below (maize, groundnuts, cassava, other roots and vegetables) have an average backward index of 0.70 and rank between 48th and 55th of 56 sectors. The same calculation on the 2013 SAM gives 0.69, so the pattern is stable. But linkage indices measure only connections between industries. They leave out the channel that matters most for poverty: income paid to households and spent again.

My economic impact analysis of Ghana's Productive Safety Net Program (PSNP) shows how much that channel matters. The PSNP is a five-year, $260 million programme in the Northern, Upper East and Upper West regions. The analysis uses the 2013 SAM for Ghana (IFPRI), with production, factors, enterprises and households all endogenous (Pyatt and Round, 1979). Each cedi spent locally generates about 1.33 cedis of value added and 1.86 cedis of household income nationally. Of the value-added effect, 76% is induced (households re-spending income), 21% is direct and only 3% comes through supply chains. Two features explain this. First, 77% of the programme's injection goes to households, workers and enterprises rather than to producers. Second, the crops that do receive money buy few inputs: value added is 86–98% of their output, which is exactly why their backward linkages are low.

The spending streams differ. Public works has the highest value-added multiplier (1.76), because most of its spending is wages, which are value added in full. Livelihood grants, mostly for staple crops, follow at 1.69. Cash transfers have the lowest value-added multiplier (1.14), because a transfer becomes value added only when it is spent, but the highest household-income multiplier (2.04). Poor farm households live on land and labour: those in the bottom quintile get 40% of their income from land, 36% from labour and 20% from capital. The PSNP figures come from the 2013 SAM rather than the 2015 SAM used above, so they illustrate the mechanism rather than offering a like-for-like comparison.

Geography matters too. Several key sectors, including petroleum refining, finance and wholesale trade, are concentrated in Greater Accra, Tema and Kumasi, and a national SAM cannot show where effects land. For the PSNP, the supply-chain effect is so small that leakage of supply-chain activity to the south barely matters: if 35–40% of it occurs outside the north, the value-added multiplier is still 1.31. What matters is where households' purchases are produced. If the same 35–40% of induced effects also leaks south, the multiplier falls to 0.91–0.96. These leakage rates are illustrative, not estimated.

Caveats

• Size. Unweighted indices treat a small sector the same as a large one. The chart sizes dots by output for this reason; output-weighted indices or the hypothetical extraction method would rank sectors by the cedi value of their linkages.
• Concentration. A high index can come from heavy dealings with a single partner. Rasmussen's coefficients of variation (lower means more evenly spread) are around 4–5 for most key sectors and around 7 for cassava, groundnuts, other roots and vegetables, whose few linkages are concentrated on a handful of partners.
• Imports. Netting out imports in proportion to each commodity's import share assumes that every user buys the same import mix, which is unlikely to hold exactly.
• The Ghosh model. As a model of how a supply shock spreads, the Ghosh model has well-known problems. Here it is used only as a descriptive measure of where each sector's output goes (Dietzenbacher, 1997). Using row sums of the Leontief inverse instead changes the list of key sectors at the margin but not the eight robust ones.
• Fixed structure. The model assumes fixed technology and prices and reflects the economy in 2015. Ghana's economy has changed since, not least through the 2022 debt crisis and the cedi's sharp depreciation.
• Linkages are not a policy prescription. Hirschman's (1958) case for prioritising key sectors leaves out employment, value added, import leakage and fiscal cost. Linkage indices are one input into those choices, not a ranking of where to invest.

All 56 sectors

#	Sector	Share of output	Backward	Forward	Quadrant	Key in all variants
1	Maize	1.1%	0.73	1.33	IV
2	Sorghum and millet	0.3%	0.69	1.34	IV
3	Rice	0.5%	0.72	1.20	IV
4	Other cereals	0.0%	0.66	0.66	I
5	Pulses	0.1%	0.75	0.75	I
6	Groundnuts	0.2%	0.71	0.70	I
7	Other oilseeds	0.5%	0.76	1.39	IV
8	Cassava	0.7%	0.70	0.67	I
9	Other roots	0.9%	0.67	0.68	I
10	Vegetables	1.4%	0.70	0.68	I
11	Sugar cane	<0.1%	1.06	1.53	III	Yes
12	Tobacco	<0.1%	0.74	1.37	IV
13	Cotton and fibers	<0.1%	0.91	1.45	IV
14	Fruits and nuts	1.3%	0.72	0.74	I
15	Cocoa	1.2%	0.73	0.68	I
16	Coffee and tea	<0.1%	0.78	0.67	I
17	Other crops	<0.1%	1.15	0.70	II
18	Cattle	0.5%	1.20	0.95	II
19	Poultry	0.5%	1.21	0.76	II
20	Other livestock	0.4%	1.18	0.78	II
21	Forestry	1.5%	1.02	1.39	III
22	Fishing	0.7%	0.78	0.85	I
23	Crude oil	4.0%	0.85	1.57	IV
24	Other mining	8.3%	1.18	0.69	II
25	Meat, fish and dairy	0.1%	1.32	0.78	II
26	Fruit and vegetable processing	<0.1%	1.00	0.72	I
27	Fats and oils	0.6%	1.24	0.84	II
28	Grain milling	2.2%	1.20	0.81	II
29	Sugar refining	<0.1%	1.39	0.83	II
30	Other foods	0.4%	1.17	0.82	II
31	Beverages	0.4%	0.99	0.84	I
32	Tobacco processing	<0.1%	1.10	0.69	II
33	Textiles	0.3%	1.10	0.87	II
34	Clothing	0.2%	1.21	1.05	III
35	Leather and footwear	<0.1%	1.03	0.70	II
36	Wood and paper	1.4%	1.43	1.24	III	Yes
37	Petroleum	3.1%	1.30	1.70	III	Yes
38	Chemicals	1.1%	0.97	1.44	IV
39	Non-metal minerals	0.5%	0.96	1.36	IV
40	Metals and metal products	0.5%	1.00	1.35	IV
41	Machinery and equipment	<0.1%	0.92	1.15	IV
42	Other manufacturing	0.1%	1.15	0.71	II
43	Electricity, gas and steam	3.4%	1.30	1.26	III	Yes
44	Water supply and sewage	1.8%	1.13	0.94	II
45	Construction	11.4%	0.80	0.71	I
46	Wholesale and retail trade	8.0%	1.17	1.21	III	Yes
47	Transportation and storage	8.9%	0.96	1.34	IV
48	Accommodation and food services	4.5%	0.94	0.73	I
49	Information and communication	3.3%	1.17	1.13	III	Yes
50	Finance and insurance	3.2%	1.02	1.82	III
51	Real estate activities	2.4%	1.02	1.38	III	Yes
52	Business services	4.3%	1.29	1.07	III	Yes
53	Public administration	5.9%	1.21	0.69	II
54	Education	3.6%	1.05	0.81	II
55	Health and social work	1.4%	1.07	0.81	II
56	Other services	2.4%	0.79	0.68	I

Domestic coefficients; backward from the Leontief inverse, forward from the Ghosh inverse. Quadrant I: below average on both; II: above-average backward only; III: above average on both (key sectors); IV: above-average forward only. "Other cereals" has no domestic output.

Data and references

GSS, ISSER and IFPRI. 2015 Nexus Social Accounting Matrix for Ghana (release of 6 May 2016). A public version is available as IFPRI (2021), 2015 Social Accounting Matrix for Ghana, Harvard Dataverse, doi:10.7910/DVN/BCPLAA.
IFPRI. 2013 Social Accounting Matrix for Ghana. Harvard Dataverse. doi:10.7910/DVN/YVZ8KR.
Dietzenbacher, E. (1997). In vindication of the Ghosh model: A reinterpretation as a price model. Journal of Regional Science, 37(4), 629–651.
Ghosh, A. (1958). Input-output approach in an allocation system. Economica, 25(97), 58–64.
Hirschman, A. O. (1958). The Strategy of Economic Development. New Haven: Yale University Press.
Miller, R. E., and Blair, P. D. (2009). Input-Output Analysis: Foundations and Extensions (2nd ed.). Cambridge: Cambridge University Press.
Pyatt, G., and Round, J. I. (1979). Accounting and fixed price multipliers in a social accounting matrix framework. The Economic Journal, 89(356), 850–873.
Rasmussen, P. N. (1956). Studies in Inter-Sectoral Relations. Amsterdam: North-Holland.