I am far too clued out about people and politics to understand the sub-text(?) behind Mark Thoma's short post. But this is a lovely exam question, that any macroeconomist ought to be able to take a crack at.
And it reminds me of my youth in the early 1970's, when Milton Friedman was arguing, against the "keynesians" (post-keynesians?), that monopoly power (either by firms or unions) did not cause inflation, which was always and everywhere a monetary phenomenon. And it reminds me of my early attempts in the 1980's to build macro models with market power, before it became commonplace in New Keynesian macro models.
So, at the risk of wandering into some sort of minefield, I can't resist answering the question.
My answer: yes/no/it depends. It depends on the shape of the AD curve, which depends on what the central bank is trying to do.
You can see the reasoning behind the idea fairly easily.
Profit-maximising firms set Marginal Revenue = Marginal Cost, and MR=(1-(1/E)).Price, so P=(1/(1-(1/E))).MC , where E is elasticity of the firm's demand curve.
Suppose a perfectly competitive firm became monopolised and less perfectly competitive. E then falls. It would then set its price as a bigger markup over marginal cost, because that markup is inversely related to the elasticity of its demand curve.
Therefore (warning bells should be ringing in your head at that "therefore"), if all firms became more monopolised, all prices should rise. And when prices rise, money wages will rise too, which means firms' marginal costs will rise, which means that prices will rise still further, which means....we get a price-wage inflationary spiral. Unless the central bank deliberately creates high enough unemployment to stop it.
Unless my memory has failed, that is how many people (economists and non-economists) used to argue back in the 1970's. And the Monetarist counter-argument, that monopoly power only affects relative prices, not money prices, and that we need to look at the demand and supply of money if we want to understand inflation, was never totally convincing. Because back then, in the days before New Keynesian macroeconomics, we didn't know how to build macroeconomic models with monopolistically competitive firms.
Despite the obvious fallacy of composition in my above reconstruction of the "keynesian" argument, and despite my own monetarist leanings, I think that argument is not wholly wrong. In fact, once you add in an accelerationist expectations-augmented Phillips Curve, it's just a different way of looking at a much more orthodox answer to the question.
Take a very simple version of a New Keynesian model. There are n firms in the economy (n is large and exogenous); each firm produces a unique product and faces a demand curve with a constant elasticity E. Let labour be the only input, with a production function y=l (and Y=L in aggregate). The labour market is perfectly competitive (to keep it simple), with a labour supply function L=S(W/P).
Let Pi be the price of firm i, and P the average of all firms' prices.
To maximise profits, the individual firm sets Pi equal to a markup over marginal cost, and since its marginal cost is the wage W:
Pi = (1/(1-(1/E))).W
Divide both sides by P, to get:
Pi/P = (1/(1-(1/E))).W/P
In symmetric Nash equilibrium all firms will set the same price, so Pi=P, which means:
1 = (1/(1-(1/E))).W/P or:
W/P = 1 - (1/E)
Plug that solution for the real wage into the labour supply function to solve for the equilibrium level of aggregate output and employment:
Y = L = S(1-(1/E))
An increase in firms' monopoly power (a fall in E) reduces real wages, employment, and output.
But what about the effect of E on the price level P? For that we need to add an Aggregate Demand curve to the model. All we have derived so far is the Long Run Aggregate Supply curve (though it isn't really a "supply" curve, because firms will happily produce and sell however much output is demanded at the prices they set).
Suppose the central bank is targeting the level of nominal GDP, so the AD curve is a downward-sloping rectangular hyperbola: P.Y=some constant. Then an increase in monopoly power (a fall in E) will cause a fall in Y and a rise in the price level. A one-time rise in the price level, not an ongoing higher rate of inflation.
Suppose instead the central bank is targeting the price level (or inflation). That makes the AD curve horizontal. If the central bank sees the fall in E and Y coming, there is no effect on the price level. If the central bank doesn't see it coming, there may be a temporary rise in P (followed by a fall in P if the central bank is targeting the price level).
But the old "keynesians" in the 1970's didn't think of monetary (and fiscal policy) that way. They thought of monetary (and/or fiscal policy) as targeting "full-employment output". They (implicitly) assumed a vertical AD. That's what they meant by "holding monetary policy constant" in the implicit "other things equal" clause of the question.
If an increase in monopoly power causes the (vertical) LRAS curve to shift to the left, so it now lies to the left of a vertical AD curve, what happens? We get ever-accelerating inflation, with the speed of the inflationary explosion limited only by the speed at which firms raise prices in response to rising wages, and wages in turn rise in response to rising prices, and the speed at which expected inflation adjusts to actual inflation. Unless the central bank shifts the AD curve left, to reduce output and employment, to stop the inflation.
But, as Friedman would note, if the Long Run Phillips Curve is vertical, because expected inflation must eventually catch up to actual inflation, you can't possibly have a long run equilibrium with stable inflation if the AD curve is vertical.
They weren't daft, those old keynesians. They just had a different implicit assumption about what "other things equal" meant. They had a different framing for monetary policy.
[Update: perhaps wandering deeper into the minefield, if you believe in liquidity traps, with a horizontal LM curve, you (normally) get a vertical AD curve. So, hmmmm. Plus, you might in fact even want (at least temporarily) an increase in monopoloy power, to increase inflation, to get the economy out of the liquidity trap, by shifting the LRAS curve left. Didn't someone build a similar model a couple of years back, where an increase in distorting taxes to reduce labour supply was one policy to get you out of the liquidity trap?]